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# In an AP, the $\; 4^{th}\;$ term is $36$. The$\; 21^{st}\;$ term is $108$ more than the $\;9^{th}\;$ term. What is the first term a and the common difference $d$?

$\begin{array}{1 1} a=15,d=12 \\ a=9,d=9 \\ a=5,d=10 \\ a=3,d=21 \end{array}$

$The\;4^{th}\;term\;a_{4}=a+3d=36$
$The\;9^{th}\;term\;a_{9}=a+8d$
$The\;21^{st}\;term\;a_{21}=a+20d=108+a+8d\;{108\;more\;than\;a_{9}}$
$Therefore\;20d-8d=108$
$12d=108$
$d=9$
$a=36-3d=36-27=9.$